We present a theoretically founded approach for high-dimensional Bayesian optimization based on low-dimensional subspace embeddings. We prove that the error in the Gaussian process model is bounded tightly when going from the original high-dimensional search domain to the low-dimensional embedding. This implies that the optimization process in the low-dimensional embedding proceeds essentially as if it were run directly on the unknown active subspace. The argument applies to a large class of algorithms and GP mod- els, including non-stationary kernels. Moreover, we provide an efficient implementation based on hashing and demonstrate empirically that this sub- space embedding achieves considerably better results than the previously proposed methods for high-dimensional BO based on Gaussian matrix projections and structure-learning.